The idea of topological T-duality is to disregard the Riemannian metric and the connection and study the remaining “topological” structure.

While the idea of T-duality originates in string theory, topological T-duality has become a field of study in pure mathematics in its own right.

In the language of bi-branes a topological T-duality transformation is a bi-brane of a special kind between the two gerbes involved. The induced pull-push operation (in groupoidification and geometric function theory) on (sheaves of sections of, or K-classes of) (twisted) vector bundles is essentially the Fourier-Mukai transformation. More on the bi-brane interpretation of (topological and non-topological) T-duality is in SarkissianSchweigert08.

Definition

Two tuples (Xi→B,Gi)i=1,2(X_i \to B, G_i)_{i = 1,2} consisting of a TnT^n-bundle XiX_i over a manifold BB and a line bundle gerbeGi→XiG_i \to X_i over XX are topological T-duals if there exists an isomorphism uu of the two line bundle gerbes pulled back to the fiber product correspondence space X1×BX2X_1 \times_B X_2:

In these papers the gerbe does not appear, but an integral 3-form, representing the Dixmier-Douady class of a gerbe does. Note that if the integralcohomology groupH3(X,ℤ)H^3(X,\mathbb{Z}) of XX has torsion in dimension three, not all gerbes will arise in this way. The formalization with the above data originates in